Problem 74
Question
Will help you prepare for the material covered in the next section. Solve: \(\frac{2 x}{3}=\frac{14}{3}-\frac{x}{2}\).
Step-by-Step Solution
Verified Answer
The solution to the equation \(\frac{2 x}{3}=\frac{14}{3}-\frac{x}{2}\) is \(x = 4\).
1Step 1: Simplify the Equation
First, the equation \(\frac{2 x}{3}=\frac{14}{3}-\frac{x}{2}\) needs to be simplified. This can be done by getting rid of the denominators by multiplying the entire equation by 6 (which is the least common multiple of 2 and 3). So, multiply every term by 6 to obtain: \(4x = 28 - 3x\)
2Step 2: Rearrange the terms
Next, rearrange the equation by moving the term with \(x\) on the right side to the left side. Add \(3x\) on both sides to get \(4x + 3x = 28\). Simplify to get \(7x = 28\)
3Step 3: Solve for \(x\)
Finally, divide both sides by 7 to isolate \(x\). This gives \(x = \frac{28}{7} = 4\)
Key Concepts
Simplifying EquationsLeast Common MultipleAlgebraic ExpressionsIsolating Variables
Simplifying Equations
When working with linear equations, one of the key initial steps is to simplify the equation. Simplifying an equation makes it easier to solve by rearranging it into a more manageable form. You can achieve this by combining like terms and removing any unnecessary complexity, such as fractions or parentheses.
In the given exercise, simplifying involves getting rid of fractions by finding a common denominator, which is done by multiplying each term by the least common multiple of the denominators. This step makes the equation more straightforward and sets the stage for easier manipulation of the terms to isolate the variable.
In the given exercise, simplifying involves getting rid of fractions by finding a common denominator, which is done by multiplying each term by the least common multiple of the denominators. This step makes the equation more straightforward and sets the stage for easier manipulation of the terms to isolate the variable.
Least Common Multiple
The least common multiple (LCM) of two numbers is the smallest number that is evenly divisible by both of them. It is a vital concept when dealing with fractions in algebraic equations.
To solve an equation with different denominators, you can multiply every term by the LCM of the denominators to eliminate the fractional terms. In our exercise, the LCM of 2 and 3 is 6. By doing this, we successfully remove the denominators, transforming the equation into a simpler form that can be solved through standard algebraic methods.
To solve an equation with different denominators, you can multiply every term by the LCM of the denominators to eliminate the fractional terms. In our exercise, the LCM of 2 and 3 is 6. By doing this, we successfully remove the denominators, transforming the equation into a simpler form that can be solved through standard algebraic methods.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. They are the building blocks of algebraic equations. In the context of the given exercise, we see the algebraic expression \(\frac{2 x}{3} = \frac{14}{3} - \frac{x}{2}\) before simplification.
Understanding how to manipulate these expressions is crucial. You can add, subtract, multiply, or divide them, just like numerical expressions, to simplify and solve equations. The idea is to perform the same operation on both sides of the equation to maintain equality, which leads us toward finding the value of the unknown variable.
Understanding how to manipulate these expressions is crucial. You can add, subtract, multiply, or divide them, just like numerical expressions, to simplify and solve equations. The idea is to perform the same operation on both sides of the equation to maintain equality, which leads us toward finding the value of the unknown variable.
Isolating Variables
Isolating the variable in an algebraic equation is a fundamental goal of solving equations. It means rearranging the equation so that the variable we're solving for is on one side of the equation, and everything else is on the other side.
In our particular problem, after simplifying the equation, we grouped all terms with the variable on one side and the constants on the other. This step often involves using addition or subtraction to move terms from one side of the equation to the other and using division or multiplication to get the variable by itself. The result is a clear solution for the variable—an essential goal for any algebraic equation you will encounter.
In our particular problem, after simplifying the equation, we grouped all terms with the variable on one side and the constants on the other. This step often involves using addition or subtraction to move terms from one side of the equation to the other and using division or multiplication to get the variable by itself. The result is a clear solution for the variable—an essential goal for any algebraic equation you will encounter.
Other exercises in this chapter
Problem 74
The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a
View solution Problem 74
Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x y-2 x}{3 y-6}$$
View solution Problem 74
Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-7}{3 y^{2}}-\frac{y-2}{12 y}$$
View solution Problem 74
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. All real numbers satisfy
View solution